The structure of the modified Bessel functions plays a key role in the evaluation of Eqs. (REF ) and (REF ) and is discussed here. The following asymptotic expansions were used in our analysis and are adjustments of those found in Abramowitz and Stegun [1]} and Wolfram Alpha [2]}:
\(I_\nu (z) \sim \frac{e^{z}}{\sqrt{2\pi z}}\mathbb {I}(\nu ,z)+i\frac{e^{-z+i\pi \nu (1-2m)-i\pi m}}{\sqrt{2\pi z}}\mathbb {J}(\nu ,z)\qquad \text{ as } |z|\rightarrow \infty ,\) 
\(m={\left\lbrace \begin{array}{ll}0, &Im(z) > 0 \\1, &Im(z) <0\end{array}\right.},\qquad \nu \in \mathbb {R},\qquad z \in \mathbb {C},\qquad z \notin \mathbb {R},\) 
\(\mathbb {I}[\nu ,z]=1-\frac{(4\nu ^2-1)}{8z}+\frac{(4\nu ^2-1)(4\nu ^2-9)}{2!(8z)^2}-\frac{(4\nu ^2-1)(4\nu ^2-9)(4\nu ^2-25)}{3!(8z)^3}\\+\frac{(4\nu ^2-1)(4\nu ^2-9)(4\nu ^2-25)(4\nu ^2-49)}{4!(8z)^4}+O\left(\frac{1}{z^5}\right),\) 
\(\mathbb {J}[\nu ,z]=1+\frac{(4\nu ^2-1)}{8z}+\frac{(4\nu ^2-1)(4\nu ^2-9)}{2!(8z)^2}+\frac{(4\nu ^2-1)(4\nu ^2-9)(4\nu ^2-25)}{3!(8z)^3}\\+\frac{(4\nu ^2-1)(4\nu ^2-9)(4\nu ^2-25)(4\nu ^2-49)}{4!(8z)^4}+O\left(\frac{1}{z^5}\right),\) 
